Closing the Gap Between Nano- and Macroscale: Atomic Interactions vs. Macroscopic Materials Behavior 11
predict a mostly linear decrease of the solution energy with increasing volumetric strain. This
a
(BCC)
-0.6
0.94
-0.4
exc
-0.2
0.0
0.2
0.4
0.6
0.8
HVH( ) - [eV]
0.96 0.98 1 1.02 1.04 1.06
VV/
exc
0
0
DFT
MEAM
Lau
Ruda
Hepburn
Fig. 5. Calculations of the strain-dependent solubility of C in an octahedral position in α-iron
(indicated as the blue atom in left panel) show that only few empirical potentials are able to
reproduce the results of corresponding DFT calculations (Hristova et al., 2011) (right panel).
can be understood intuitively in terms of the additional volume of the supercell that can be
accommodated by the carbon atom. Despite this comparably simple intuitive picture, the
majority of investigated EAM and MEAM potentials deviate noticeably from the DFT results.

, . . . and the discrete displacements ξ
α
, ξ
β
, ξ
γ
, ,namelyX
α
=
X
α
0
+ ξ
α
, X
β
= X
β
0
+ ξ
β
, . . . (c.f., Figure 6). By introducing the distance vectors:
R
αβ
0
= X
β
0
−X
α

β
0
)=u(X
0
)+
∂u
∂X
0
·R
αβ
0
+ , (23)
R
αβ
= R
αβ
0
+
∂u
∂X
0
·R
αβ
0
= F · R
αβ
0
. (24)
Here the symbol F
= I +

)+
∑
β
(α=β)
∂E
α
∂R
αβ



R
αβ
0
·

R
αβ
−R
αβ
0

+
+
1
2
∑
β
(α=β)
∂

X
1
X
2
X
3
X
α
0
X
α
0
X
α
X
α
X
β
0
X
β
0
X
β
X
β
undeformed state
deformed state
zoomed view
a

u)
T
)=E. Substituting R
αβ
−R
αβ
0
by Eq. (24) yields:
E
α
(R
α1
, ,R
αN
)=E
α
(R
α1
0
, ,R
αN
0
)+E ·
∑
β
(α=β)
∂E
α
∂R
αβ



R
αβ
0
R
αβ
0

··E
T
. (26)
140
Thermodynamics – Kinetics of Dynamic Systems
Closing the Gap Between Nano- and Macroscale: Atomic Interactions vs. Macroscopic Materials Behavior 13
An alternative formulation of E
α
is given by considering the scalar product of the atomic
distance vector R
αβ
,viz.
R
αβ 2
= R
αβ
·R
αβ
=(F ·R
αβ
0

2
)=E
α
(R
α1
0
2
, ,R
αN
0
2
)+2G ··
∑
β
(α=β)
E
α

R
αβ
0
R
αβ
0
+
+
4
2
G
··

αβ2
=R
αβ2
0
. Since first derivatives of the energy must
vanish for equilibrium (minimum of energy) this expression allows to directly identify the
equilibrium condition, which - in turn - provides an equation for calculating the lattice
parameter a. Furthermore the last term of Eq. (28) can be linked to the stiffness matrix
C
=[C
ijkl
], which contains the elastic constants of the solid. However, the atomic energy
E
α
in Eq. (28) must be formulated in terms of the square of the scalar distances R
αβ
between
the atoms α, β
= 1, ,N.
3.2 Brief survey of JOHNSON’s analytical embedded-atom method
The specific form of E
α
, E
α

and E
α

in Eq. (28) strongly depends on the chosen interaction
model and the corresponding parametrization, i.e., the chosen form of the function(s), which

(R
2
)=ρ
(e)
exp

− β

R
2
R
2
0
−1

, φ
AA
(R
2
)=φ
(e)
exp

−γ

R
2
R
2
0

14 Will-be-set-by-IN-TECH
Originally, JOHNSON used the scalar distance R within the above equations, but due to
the explanations in Section 3.1 the present formulation in terms of R
2
is used by simple
substitution (Böhme et al., 2007). By using the universal equation of state derived by R
OSE
and COWORKERS (Rose et al., 1984) the embedding function reads:
F
A
(ρ
A
)=−E
sub

1
+ α


1 −
1
β
ln
¯
ρ
A
¯
ρ
(e)
A


γ
β
(30)
with α
=

κΩ
(e)
/E
sub
;(Ω
(e)
: volume per atom). Hence three functions φ
AA
, ρ
A
,
and F
A
must be specified for the pure substance "A", which is done by fitting the five
parameters α, β, γ, φ
(e)
, ρ
(e)
to experimental data such as bulk modulus κ,shearmodulus
G , unrelaxed vacancy formation energy E
u
v
, and sublimation energy E

ρ
A
φ
AA
+
ρ
A
ρ
B
φ
AA

. (31)
Consequently all functions are calculated from information of the pure substances; however
10 parameter must be fitted. In Figure 8 the different functions according to Eq. (19) are
illustrated for both FCC-metals Ag and Cu. The experimental data used to fit the EAM
parameters are shown in Table 1.
atom a in Å E
sub
in eV E
u
v
in eV κ in eV/Å
3
G in eV/Å
3
Ag 4.09 2.85 1.10 0.65 0.21
Cu 3.61 3.54 1.30 0.86 0.34
Table 1. Experimental data for silver and copper (the volume occupied by a single atom is
calculated via Ω

derived in Section 3.1 and expand
φ
αβ
(R
αβ 2
), ρ
β
(R
αβ 2
) as well as F
α
(
∑
ρ
β
(R
αβ 2
)) around R
αβ 2
0
. Then the energy of atom α reads:
E
α
=
1
2
∑
β
φ
αβ

α
+ 2F

α
(
¯
ρ
(e)
α
)W
α
+ 2F

α
(
¯
ρ
(e)
α
)V
α
V
α

··G (32)
in which the following abbreviations hold:
A
α
=
∑

0
R
αβ
0
,
V
α
=
∑
β
ρ

β
(R
αβ 2
0
)R
αβ
0
R
αβ
0
W
α
=
∑
β
ρ

β

αβ 2
0
)=6φ
(e)
and
¯
ρ
(e)
α
= 12ρ
(e)
α
.
Three parts of Eq. (33) are worth-mentioning: The first two terms represent the energy of
atom α within an undeformed lattice. The term within the brackets
[ ] of the third summand
denotes the slope of the energy curves in Figure 8 (a). If lattice dynamics is neglected, the
relation A
α
+ 2F

α
(
¯
ρ
(e)
α
)V
α
= 0 will identify the equilibrium condition and defines the nearest

+ 2F

α
(
¯
ρ
(e)
α
)V
α
V
α
].
Thus, in case of the above analyzed metals Ag and Cu, we obtain the following atomistically
calculated values
6
(for comparison the literature values (Kittel, 1973; Leibfried, 1955) are
additionally noted within the parenthesis):
C
Ag
1111
= 132.6 (124) GPa , C
Ag
1122
= 90.2 (94) GPa , C
Ag
2323
= 42.4 (46) GPa ,
C
Cu

.
Obviously the discrepancy between the theoretical calculations and experimental findings is
6
There are three non-equivalent elastic constants for cubic crystals (Leibfried, 1955).
143
Closing the Gap Between Nano- and Macroscale:
Atomic Interactions vs. Macroscopic Materials Behavior
16 Will-be-set-by-IN-TECH
R
(a) energy of atom α
()
(b) embbeding function
R
(c) atomic electronic density
R
(d) pairwise, repulsive interaction
Fig. 8. Different contributions to the EAM potential for silver (α = 5.92, β = 2.98, γ = 4.13,
φ
(e)
= 0.48 eV/Å
3
, ρ
(e)
= 0.17 eV/Å
3
) and copper (α = 5.08, β = 2.92, γ = 4.00, φ
(e)
= 0.59
eV/Å
3

is the KRONECKER symbol. Then φ
αβ
and
¯
ρ
(e)
α
can be written as:
φ
αβ
= φ
AA
+

ˆ
y
α
+(1 −2
ˆ
y
α
)
ˆ
y
β

φ
+(
ˆ
y

with the definitions φ = φ
AB
−
1
2
(φ
AA
+ φ
BB
) and
˜
φ =
1
2
(φ
BB
− φ
AA
).Here
ˆ
y
γ
acts as a
"selector", which provides the corresponding interaction terms depending on which pair of
atoms is considered. Thus, in particular,
ˆ
y
α/β
are both zero, if two "A" atoms are considered
and φ

∂y
∂X
0
·R
αβ
0
+
1
2
∂
2
y
∂X
2
··R
αβ
0
R
αβ
0
(36)
yields the so-called mean-field limit
7
,viz.
φ
αβ
= φ
AA
+ 2y(1 − y )φ + 2y
˜

∑
β
(
¯
ρ
B
−
¯
ρ
A
) . (38)
In a similar manner the embedding function F
α
in Eq. (32) is decomposed:
F
α
(
¯
ρ
(e)
α
)=(1 −y)F
A
+ yF
B
, (39)
but note that the argument of F
A/B
is also defined by a decomposition according to Eq. (38).
Therefore F

2
y).
Moreover, the quantities A
α
, B
α
, F

α
V
α
, F

α
V
α
V
α
,andF

α
W
α
can be also treated analogously
to Eqs. (39-37). Finally, one obtains for the energy of an atom α within a binary alloy, see also
(Böhme et al., 2007) for a detailed derivation:
E
α
(y)=
1

Δ

F

A
+ y(F

B
− F

A
)


+
+
1
2
G
··

2B
A
+ 4yB
˜
φ
+ 2y(1 − y )B
φ
+ 4



F

A
+ y(F

B
− F

A
)


··G + O(∇y, ∇
2
y) (40)
with the abbreviations: g
AA
=
∑
β
φ
AA
, g
φ
=
∑
β
φ, g
˜

or φ

;
˜
φ

or
˜
φ

,and
(ρ

B
−ρ

A
) or (ρ

B
−ρ

A
). Eq. (40) indicates various important conclusions:
• The terms of the first row stand for the energy of the undeformed lattice. Here no
mechanical effects contributes to the energy of the (homogeneous) solid. These energy
7
For homogeneous mixtures concentration gradients can be neglected; for mixtures with spatially
varying composition terms with
∇y = ∂y/∂X

A
+ yV
Δ

F

A
+ y(F

B
− F

A
)

≡ 0 (41)
yields a
(e)
(y) as a function of the concentration, cf. example below.
• The term of the third and last row denotes the elastic energy E
elast
=
1
2
E ··C(y) ··E with
E ≈ G of an atom in the lattice system. Consequently, the bracket term characterizes the
stiffness matrix of the solid mixture, viz.
C
(y)=
1

+
+
4

V
A
+ yV
Δ

V
A
+ yV
Δ

F

A
+ y(F

B
− F

A
)


. (42)
Note that Ω
(e)
(y) is calculated by a

y = 0.1
0.3
0.5
0.7
0.9
R
Fig. 9. Left: Left hand side of the equilibrium condition for different, exemplarily chosen
concentrations (R
2
0,Ag
= 8.35, R
2
0,Cu
= 6.53). Right: Calculated equilibrium lattice parameter
as a function of concentration.
The three independent elastic constants for the mixture Ag-Cu are calculated by Eq. (42) and
illustrated in Figure 10. Here we used a
(e)
(y
i
),withy
i
= 0, 0.05, . . . , 0.95, 1 correspondingly to
Figure 9 (right). It is easy to see, that for y
= 0(Ag)andy = 1 (Cu) the elastic constants of
146
Thermodynamics – Kinetics of Dynamic Systems
Closing the Gap Between Nano- and Macroscale: Atomic Interactions vs. Macroscopic Materials Behavior 19
silver and copper, illustrated on page 15, result. However, for 0 < y < 1 the elements of the
stiffness matrix do not follow the linear interpolation as indicated in Figure 10.

φ
(y)+G(y) ··B
φ
··G . (44)
Here g
φ
as well as B
φ
are given by the interatomic potentials
8
and must be evaluated at the
concentration dependent nearest neighbor distance R
0
(y)=a
(e)
(y)/
√
2, which - in turn -
follows from the equilibrium condition. Thus, symmetry of Eq. (43) does not necessarily exist.
Moreover, further investigations of Eq. (44) may allow a deeper understanding of non-ideal
energy-contributions to solid (and mechanically stressed) mixtures.
8
Note, that Λ exclusively depends on the pairwise interaction terms; contributions from the embedding
functions naturally cancel.
147
Closing the Gap Between Nano- and Macroscale:
Atomic Interactions vs. Macroscopic Materials Behavior
20 Will-be-set-by-IN-TECH
4. Thermodynamic properties
Atomistic approaches for calculating interaction energies cannot only be used to quantify

B
) ln(1 −y
B
)

+
˜
g
exc
(y, T) .
(45)
The first and second term represent the contributions from the pure substances; the third
summand denotes the entropic part of an ideal mixture
−T
˜
s(y)=−N
A
k
B
T
∑
2
i
=1
y
i
ln y
i
with
N

BB
+ F
B
)+k
B
T

y
B
ln y
B
+(1 −y
B
) ln(1 −y
B
)

+
+
12y(1 −y)φ . (46)
Obviously, the G
IBBS free energy curve is superposed by three, characteristic parts, namely
(a) a linear function interpolating the energy of the pure substances; (b) a convex, symmetric
entropic part, which has the minimum at y
= 0.5 and vanishes for y = {0, 1} and (c) an excess
term, which - in case of binary solids with miscibility gap - has a positive, concave curve
shape, cf. Figure 11 (right). Hence, a double-well function results, as illustrated in Figure 11
(left) for the cases of Ag-Cu at 1000 K. Here the concave domain y
∈ [0.19, 0.79] identifies the
unstable regime, in which any homogeneous mixture starts to decompose into two different

=
g(y
(β)
, T) − g(y
(α)
, T)
y
(β)
−y
(α)
. (47)
Eq. (47) provides two equations for the two unknown variables y
(α)/(β)
. The quantity g(y, T)
as well as its derivatives can be directly calculated from the atomistic energy expression in
Eq. (46).
Figure 12 (squared points) displays the calculated equilibrium concentrations for different
temperatures. Here the dashed lines represent experimental data adopted from the database
MTData
TM
. As one can easily see, there is good agreement between the experimental
148
Thermodynamics – Kinetics of Dynamic Systems
Closing the Gap Between Nano- and Macroscale: Atomic Interactions vs. Macroscopic Materials Behavior 21
T
p
= 1000 K
= 1 bar
MTDATA
TM

thus, temperature-depending materials properties can only be precisely determined on the
atomistic scale by incorporating lattice vibrations, i.e. phonons.
To this end the lattice is modelled as a 3D-many-body-system, consisting of mass points
(atoms) and springs (characterized by interatomic forces). Thus, the equation of motion
of atom α can easily be found by the framework of classical mechanics. By considering
m
α
¨
ξ
α
= F
α
= −∇E
α
and Eq. (25) one can write the following equation of motion for the
149
Closing the Gap Between Nano- and Macroscale:
Atomic Interactions vs. Macroscopic Materials Behavior
22 Will-be-set-by-IN-TECH
700
800
900
1000
1100
1200
1300
1400
0 0,2 0,4 0,6 0,8 1
y
Cu

∂
2
E
α
∂R
αβ
∂R
αβ



R
αβ
0
·(R
αβ
−R
αβ
0
) . (48)
In what follows we restrict ourselves to the so-called harmonic approximation, which means
that terms beyond quadratic order are neglected in Eq. (25). Please note the identity R
αβ
−
R
αβ
0
= ξ
β
− ξ

∂
2
E
α
∂R
αβ
∂R
αβ



R
αβ
0
·e

1
−e
ik·R
αβ
0

=
∑
β
D
αβ
(R
αβ
0

αβ 2
, respectively.
Therefore the chain rule ∂
2
E
α
/(∂R
αβ
)
2
=(∂
2
E
α
/∂x
2
)(∂x/∂R
αβ
)
2
+(∂E
α
/∂x)(∂
2
x/∂R
αβ 2
)
must be applied to obtain D
αβ
.

αβ
−Im
α
ω
2

= 0 . (51)
The three eigenvalues,
˚
D
I/II/III
(k)=m
α
ω
2
I/II/III
(k),ofthe3×3 matrix
˚
D
αβ
yield the
eigenfrequencies ν
I/II/III
(k)=ω
I/II/III
(k)/(2π). Additionally, Eq. (51) defines three
eigenvectors e
I/II/III
with e
k

literature, (Bian et al., 2008; Svensson et al., 1967).
151
Closing the Gap Between Nano- and Macroscale:
Atomic Interactions vs. Macroscopic Materials Behavior
24 Will-be-set-by-IN-TECH
At finite temperature real crystal vibrations show a wide range of wave vectors and
frequencies. To quantify the dynamical characteristics of the lattice phonon dispersion curves
are measured (or calculated), which displays all frequencies for the lattice-specific symmetry
directions. Figure 13 illustrates the phonon dispersion curves, calculated from the atomistic
model for copper. Here we considered the three elemental symmetry directions of the
FCC-structure, namely ξ
[100], ξ[011],andξ[111] with ξ ∈ [0, 2π/a
(e)
] or ξ ∈ [0, π/a
(e)
],
respectively (1st B
RILLOUIN zone
9
). The squared, discrete points are added for comparative
purposes and identify experimental data obtained from (Bian et al., 2008; Svensson et al.,
1967).
By means of quantum-mechanics and statistical physics the kinetic energy, resulting from
lattice vibrations, can be written as, cf. (Leibfried, 1955):
E
α
kin
(T)=
1
N

, (52)
in which the variable h
= 6.626 · 10
−34
Js denotes PLANCK’s constant. Furthermore the
summation is performed over all occurring eigenfrequencies ν
1
, ,ν
3N
of the N atoms within
the lattice system and the wave vectors k
. The relation of Eq. (52) results from considering
the 6N-dimensional phase space, well-established in statistical mechanics, and by adding the
energy-contribution of each oscillator to the partition function Z. Consequently an expression
for the total kinetic energy E
tot
kin
is obtained, from which E
α
kin
follows by introducing the factor
1/N. The total energy of atom α can now be written as:
E
α
tot
(T, y)=E
α
(EAM)
(y)+E
α

tot
(T, y)
∂T
=
dE
α
kin
(T)
dT
. (54)
Figure 14 compares the calculated heat capacity for copper with the experimental one
constructed from the measured E
INSTEIN frequency and the homonymous ansatz for c
v
,
(Fornasini et al., 2004).
9
The first BRILLOUIN zone represents the unit cell in the reciprocal lattice, for more details see for
example (Yu & Cardona, 2010).
10
This condition can be guaranteed by setting e.g. a = a
(e)
but any volume-preserving deformation is
possible.
152
Thermodynamics – Kinetics of Dynamic Systems
Closing the Gap Between Nano- and Macroscale: Atomic Interactions vs. Macroscopic Materials Behavior 25
fit of experimental data atomistic theory
EINSTEIN’s model
c

/h = 4.96 THz measured by Extended X-ray-Absorption Fine-Structure
(EXAFS), (Fornasini et al., 2004).
Due to the vibrational part the total energy of atom α additionally depends on T.Thethermal
expansion coefficient can be calculated by expanding E
α
tot
into a TAYLOR series according to
Eqs. (26,28), but additionally incorporating derivatives of T. A subsequent exploitation of
terms of mixed derivatives yields the thermal expansion coefficient, cf. (Leibfried, 1955), pp.
235 ff
An alternative approach for the determination of the thermal expansion coefficient is given
by the following illustrative arguments, see also Figure 15 (upper left). For T
= 0 atom
α is situated in the potential energy minimum defined by the equilibrium nearest neighbor
distance R
0
.ForT > 0 the atoms oscillates around the equilibrium position. Here the sum
E
pot
+ E
kin
defines the oscillating distance R
−
and R
+
, cf. Figure 15 (upper left). The center
position R
01
= R
−

= α
th
(T)( T − T
ref
) , α
th
(T)=α
th
(T) I , α
th
(T)=
1
R
01
(T)
dR
01
(T)
dT
. (55)
An exploitation of Eq. (55) at T
= 300 K yields the thermal expansion coefficient of α
th
=
9.1 · 10
−6
K
−1
. This value is smaller than the corresponding literature value α
th

, can be easily
calculated via the relation c
p
(T)=c
v
(T)+3T(C
1111
+ 2C
1122
)α
2
th
(T). Finally we emphasize,
that the above framework can be also applied to solid mixtures. For this reason the dynamical
matrix
˜
D
αβ
(k, y) must be calculated by the first line of the energy expression in Eq. (40).
Please note the additional argument y in
˜
D
αβ
, and consequently in ν
i
(k, y) and E
α
kin
(T, y).
Furthermore one needs the mean field relation m

−
)
R
+
− R
−
0 [K]
150
300
T=
500
700
a1
E*
R
01
Fig. 15. Upper left: On the origin of thermal expansion. The shift from R
αβ
0
to R
αβ
01
results from
the asymmetric energy curve around the minimum. Upper right: kinetic energy of Cu-atoms
calculated for different temperatures. Lower left: E
∗
(R, T)=E
α
pot
(R)+[E

textbooks (Allen & Tildesley, 1989; Frenkel & Smit, 2001).
The starting point of an MD-simulation is the choice of a thermodynamic ensemble
that determines which thermodynamic variables are conserved during the runtime of the
simulation. The thermodynamic variables most relevant for applications are temperature T,
pressure P,volumeV, internal energy E, particle number N and chemical potential μ.The
most important ensembles for MD simulations are
• the microcanonical ensemble with constant N, V, E,
• the canonical ensemble with constant N, V, T,and
• the grand-canonical ensemble with constant μ, V, T.
These macroscopic thermodynamic variables are implicitly included in an atomistic
simulation. Their calculation provides a direct link between the macroscopic (system-wide)
properties and the microscopic (atom-resolved) MD-simulation. For example, the
system-wide instantaneous temperature at a time t is calculated by equipartitioning the kinetic
energy of N atoms
1
2
k
B
T(t)=
N
∑
α=1
1
2
m
α
[v
α
(t) · v
α

0
50
100
time [fs]
-4
-3
-2
-1
0
1
energy [eV/atom]
0
1600
T [K]
temperature
kinetic energy
total energy
potential energy
Fig. 16. Time evolution of temperature and energy contributions in an MD-simulation that
employs an NVE ensemble.
max
X
a
X
a
F
a
(X
a
)

δt
2
+ O(δt
4
) (57)
as MD-integrator scheme with an error of the order of δt
4
. Due to the absence of
velocities in the extrapolation of positions the V
ERLET algorithm cannot be coupled with
thermostats/barostats and hence is suitable for NV E ensembles only. Other ensembles can
be realised with, e.g., the
VELOCITY-VERLET algorithm that involves both, positions and
velocities
X
α
(t + δt)=X
α
(t)+v
α
(t)δt +
1
2
a
α
(t)δt
2
+ O(δt
4
) , (58)

measures the correlation between the
probabilities ρ
(X
α
) and ρ(X
α,∗
) of finding atom β in an infinitesimal volume element at X
α
or
X
α,∗
, respectively, and the probability ρ(X
α
, X
α,∗
) of finding atoms in both volume elements.
ρ
(X
α
, X
α,∗
)=
[
ρ(X
α
)ρ(X
α,∗
)
]
g

Atomic Interactions vs. Macroscopic Materials Behavior
30 Will-be-set-by-IN-TECH
Another indicator in this direction is the mean square displacement
∂
∂t

ξ
(t)
2

=
∂
∂t
⎡
⎣
1
N
N
∑
β=1
ξ
β
(t)
2
⎤
⎦
= 6D , (61)
that relates the microscopic displacements, ξ
β
= X

interatomic interactions is carried out with an embedded-atom potential described earlier.
The parametrisation of the EAM potential was particularly optimised for the description of
the undercoordinated atoms at the grain boundary (Hammerschmidt et al., 2005). The atomic
structure shown in Figure 19 was obtained by (i) determining the energetically favored atomic
structure of the Σ7
(0001) coincidence-site lattice (CSL) grain boundary, (ii) setting up a block
of CSL cells of orientation
A surrounded by cells of orientation B and (iii) relaxing the atoms
in the interface area. The atomic relaxation of the interface region did not allow the grain
to decay, but resulted in a change of the grain shape from rectangular to nearly circular.
Visualising the relaxed grain (Figure 19) along the crystal axis [1000] in Figure 20 allows one
to easily distinguish the misoriented grain from the surrounding. In order to investigate the
Fig. 20. Initial isolated grain (a), viewed along [0001], undergoes a structural transformation
and orients itself to match the surrounding crystal directions (b). The MD-simulation was
carried out for 20 ps at 300 K with the bottom three layers fixed, cf. (Hammerschmidt et al.,
2005).
structural stability of the isolated grain at elevated temperatures, we carried out molecular
dynamic simulations. In particular, we simulated an NVT ensemble for 20 ps at 300 K where
we kept the bottom three layers fixed in order to mimic a microstructured substrate. The
central finding of this simulation is the decay of the isolated grain within a very short time
already at room temperature. Repeating this procedure for isolated grains of different sizes
showed that the thermal stability increases with diameter. In particular, we found that grains
with a diameter of at least 33Å are thermally stable over a maximum simulation time of
several hundred ps. This compares well with the experimentally observed minimum grain
size.
In this example, the analysis of the molecular dynamic simulation is straight-forward.
However, simulations of long times and/or large systems make it hard to identify particular
events due to the shear mass of data on atomic trajectories. This calls for approaches that
transform the information on atomic positions to meaningful derived atom-based properties
like e.g. the moments of the bond-order potentials, Eq. (13), or to even coarse-grained entities

boundaries in steel under the allowance of different alloying elements, (Nazarov et.al.,
2010), and investigations of the influence of hydrogen on the elastic properties of α-iron,
(Psiachos et al., 2011).
Nevertheless, the bridging of length- and timescales is still a big challenge for most
multiscale approaches. Here information of the nano- (e.g. binding energies of different
H-traps, such as dislocations and phase boundaries) and microscale (e.g. the temporal and
spatial phase distribution in multiphase materials) must be incorporated in macroscopic,
constitutive equations (e.g. the diffusion equation with source/sink-term for hydrogen
trapping, (McNabb & Foster, 1963; Oriani, 1970)). Moreover, the ongoing increase of
computational capacities and the development of suitable interfaces for considering atomistic
or microstructural calculations in commercial simulation software will further establish
multiscale approaches in materials engineering. The FE
2
-method, for instance, described
by (Balzani et.al., 2010) shows the large potential for incorporating micro- or mesoscopic
information in macroscopic simulations, but also the need for further acceleration of
numerical calculations and the development of optimized algorithms.
7. References
Abell, G.C. (1985). Empirical chemical pseudopotential theory of molecular and metallic
bonding, Phys. Rev. B, Vol. 31, 6184–6196 (and references therein).
Alinaghian, P.; Gumbsch, P.; Skinner, A.J. & Pettifor, D.G. (1993). Bond order potentials: a
study of s- and sp-valent systems, J. Phys.: Cond. Mat., Vol. 5, 5795–5810.
160
Thermodynamics – Kinetics of Dynamic Systems
Closing the Gap Between Nano- and Macroscale: Atomic Interactions vs. Macroscopic Materials Behavior 33
Allen, M.P. & Tildesley, D.J. (1989). Computer simulation of liquids, Oxford University Press,
ISBN: 019-8-55645-4, New York.
Aoki, M.; Nguyen-Manh, D.; Pettifor, D.G. & Vitek, V. (2007). Atom-based bond-order
potentials for modelling mechanical properties of metals. Prog. Mat. Sci., Vol. 52,
154–195.

Born, M. & Oppenheimer, R. (1927). Zur Quantentheorie der Molekeln, Ann. Phys., Vol. 84,
457–484.
Cahn, J.W. (1968). Spinodal decomposition. Trans. Metall. Soc. AIME, Vol. 242, 166–180.
Cryot-Lackmann, F. (1967). On the electronic structure of liquid transition metals. Adv. Phys.,
Vol. 16, 393–400.
Daw, M.S. & Baskes, M.I. (1983). Semiempirical, quantum mechanical calculation of hydrogen
embrittlement in metals. Phys.Rev.Lett., Vol. 50, 1285–1288.
Daw, M.S. & Baskes, M.I. (1984). Embedded-atom method: derivation and application to
impurities, surfaces and other defects in metals. Phys. Rev. B, Vol. 29, 6443–6453.
De Fontaine, D. (1975). Clustering effects in solid solutions, In: Treatise in solid state chemistry,
Vol. 5: Changes of state, N. B. Hannay (Ed.), 129-178, Plenum, New-York.
Desai, S.K.; Neeraj, T. & Gordon, P.A. (2010). Atomistic mechanism of hydrogen trapping in
bcc Fe-Y solid solution: A first principles study, Acta Mater., Vol. 58, 5363–5369.
161
Closing the Gap Between Nano- and Macroscale:
Atomic Interactions vs. Macroscopic Materials Behavior
34 Will-be-set-by-IN-TECH
Chiu, J T.; Lin, Y Y.; Shen, C L.; Lo, S P. & Wu, W J. (2008). Atomic-scale finite-element
model of tension in nanoscale thin film, Int. J. Adv. Manuf. Technol., Vol. 37, 76–82.
Drautz R. & Pettifor, D.G. (2006). Valence-dependent analytic bond-order potential for
transition metals, Phys. Rev. B, Vol. 74, 174117/1–174117/14.
Drautz, R.; Zhou, X.W.; Murdick, D.A.; Gillespie, B.; Wadley, H.N.G. & Pettifor, D.G. (2007).
Analytic bond-order potentials for modelling the growth of semiconductor thin films,
Prog. Mat. Sci., Vol. 52, 196–229.
Dreizler, R.M. & Gross, E.K.U. (1990). Density Functional Theory, Springer, ISBN: 3-540-51993-9,
Berlin.
Feraoun, H.; Aourag, H.; Grosdidier, T.; Klein, D. & Coddet, C. (2001). Development of
modified embedded atom potentials for the Cu-Ag system. Superlattice Microst.,Vol.
30, No. 5, 261–271.
Finnis, M.W. & Sinclair, J.E. (1984). A simple empirical N-body potential for transition metals,

volumetric strain and close to the Σ5(310)[001] grain boundary: Comparison of DFT
and empirical potentials, Comp. Mat. S ci., Vol. 50, 1088–1096.
Johnson, R.A. (1972). Relationship between two-body interatomic potentials in a lattice model
and elastic constants. Phys. Rev. B, Vol. 6, 2094–2100.
162
Thermodynamics – Kinetics of Dynamic Systems
Closing the Gap Between Nano- and Macroscale: Atomic Interactions vs. Macroscopic Materials Behavior 35
Johnson, R.A. (1974). Relationship between two-body interatomic potentials in a lattice model
and elastic constants II. Phys. Rev. B, Vol. 9, 1304–1308.
Johnson, R.A. (1988). Analytic nearest-neighbor model for FCC metals. Phys.Rev.B, Vol. 37,
3924–3931.
Johnson, R.A. (1989). Alloy model with the embedded-atom-method. Phys. Rev. B, Vol. 39,
12554–12559.
Jones, J.E. (1924). On the Determination of Molecular Fields. II. From the Equation of State of
aGas,Proc. R. Soc. Lond. A, Vol. 106, No. 738, 463–477.
Kadau, K.; German, T.C.; Lomdahl, P.S.; Holian, B.L.; Kadau, D.; Entel, P.; Kreth,
M.; Westerhoff, F. & Wolf, D.E. (2004). Molecular-Dynamics Study of Mechanical
Deformation in Nano-Crystalline Aluminum, Metall. Mater. Trans. A, Vol. 35A,
2719–2723.
Kagaya, H M.; Shoji, N. & Soma, T. (1988). Anharmonic effects on the thermal properties of
Si and Ge. Solid State Commun., Vol. 65, No. 11, 1445–1450.
Kittel, C. (1973). Einführung in die Festkörperphysik (3. erweiterte und verbesserte Auflage), R.
Oldenbourg Verlag GmbH München Wien, John Wiley & Sons GmbH, Frankfurt,
ISBN: 978-3-486-32763-2.
Kobayashi, R.; Warren, J.A. & Carter, W.C. (2000). A continuum model of grain boundaries.
Physica D, Vol. 140, 141–150.
Kohn, W. & Sham, L.J. (1965). Self-consistent equations including exchange and correlation
effects, Phys. Rev., Vol. 140, A1133–A1138.
Kohn, W. (1998). An essay on condensed matter physics in the twentieth century, Rev. Mod.
Phys., Vol. 71, S59–S77.